Gabriel's Horn - Mathematical Paradox
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A 3D representation of the equation y=1/x which creates mathematical paradox.
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Model information
- Source
- Printables
- Category
- Math
- License
- Creative Commons — Public Domain
This is a model of the mathematical object known as Gabriel's Horn (Wikipedia article, Numberphile video). It is a revolution around the x-axis of the equation y=1/x, from x=1 to infinity. Obviously my mk3s+ does not have an infinite z-axis height so it has to be cut off, but just imagine it continuing to infinity never quite coming to a closed point. For this scale print the next equal length of the horn looks essentially like a straight tube so I have not included any "extensions" .
The paradox is that for the purely mathematical object, the volume is finite (and actually equal to pi) but the surface area is infinite. So, theoretically, you could pour pi gallons of paint into the horn and it would be completely filled, but the inside surface would still require an infinite amount of paint to cover it. Even though in the real world filling the horn should have already coated the inside fully. A mind-bending paradox!